First part
100
Five hundred, cut in two. Build the table by hand — the way it was done before anyone thought to draw it.
Click anywhere on the rope, drag the knot, or type a cut. The rope is 500 long.
| cut | first | second | product | d | d² | 62500 − d² |
|---|
Nothing recorded yet. Record six or more cuts and the ledger will start telling you things.
Locked until the ledger holds six cuts, at least one of them on each side of the middle.
Stop naming the two parts. Name instead how far the cut sits from the middle. Write the cut as 250 + d, so the parts are 250+d and 250-d. Their sum is 500 whatever d is. Three columns have appeared in your ledger.
The last column is not a new measurement. It is 62500 - d^2, computed from your d. Compare it against the product column you filled in by multiplying.
Now read the middle column. Every entry of d^2 is zero or positive — it is a square. So every product in your ledger is 62500 with something non-negative taken away. Nothing you can write in the cut box will ever get you above 62500, and the only way to reach it is d = 0.
That is a proof of a maximum. You did not find it by trying values; the values were only how you noticed. It is true because a square is never negative.
Name a product and the ledger will tell you which cuts give it — or that none does.
quad-06-the-turn. Fill the ledger first.